Category: multiplication

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There seems to be a big “fight” about “which should you teach first, math skills or math concepts.” A popular example is the “multiplication tables” versus the concept of multiplication (as repeated addition, for example).

It’s a pretty good bet to say that when memorizing things it’s easier if you can relate the objects. Like if you went shopping and had to get toothpaste, a toothbrush and dental floss, that would be easier to remember than if you had to get shoe polish, armadillo meat and an f-string for a lute (do lutes even have f-strings?)

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Math Mojo has got some surprises for you. New lessons on how to improve your basic math skills, and videos! Professor Homunculus is getting his Video Mojo workin’ to bring you some great new stuff.

The first set of videos will be about how to practice multiplication using playing cards. So grab a deck of cards and let’s get going!

First, take out all the Spade cards from the deck - we’ll only be using those. Then, remove the court cards (the Jacks, Queens and Kings) from those cards. Consider the Ten to be a zero and the Ace to be a one.

Now you’ve got 10 cards, which represent the digits zero through nine.

Shuffle the cards. Now decide, in your mind, which digit you’d like to multiply by.

Deal the cards, face up, on the table so that you can see the faces of all the cards.

Get out a piece of paper and a pencil.

Depending on how advanced you are at multiplication, start at either the right (if you multiply the “school” way) or the left (if you know Math Mojo) of the spread deck, and start multiplying, writing only the answer (not the carries - never write the carries!)

In the video, we’ll be multiplying all the digits from 0 to 9, by 3. It’s simple to start with 3.

After you learn how to do it, try multiplying the cards by the other digits.

We’ll multiply by some higher digits in future videos.

You may have noticed that I don’t know my left from my right in this video. My bad!

Tomorrow we’ll practice checking, using this same example.

Today a concerned reader took issue with what he understands my methods to be. (See comment #4 at Augends, Addends and the Commutative Law of Addition.)
Fair enough, but I think he may have misunderstood my methods.

That could, of course, be due to the way I communicate them (or miscommunicate them). First let me say that none of the algorithms (ways of solving math problems) I teach are “mine.” “Math Mojo” is the name of my attitude, not the methods. The methods have been either gleaned from better sources than me (and most are hundreds, if not thousands, of years older), or I have “re-invented” them. That is typical for most people’s alternative methods.

Now to the issue; the reader stated:

After all these years (30) of struggling to teach children math, I finally realize why it is so difficult. A brief perusal of some of the mathematical girations you go through to multiply two numbers together explains a lot of why kids are poor at math. Commutative and associative properties are more easily understood when you have the basic tools to work with without adding zeros then subtracting the number from your cousins name on your mother’s side of the family. Teach the basics by rote then progress to the more abstract. Simple to complex seems to work.

Professor Homunculus’ reply:

I’m sorry you’ve come to that conclusion. If you’ve been teaching math for 30 years, you surely have some insights. But I can’t see see how you’d say, “simple to complex” seems to work. May I ask where it seems to work? And if it does, why is it a struggle for you, and why is it so difficult? Have you been teaching with the “girations” (sic) you say I use to make it so frustrating?

I’m not quite sure I understand the logic of your position.

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Recently I got an request to review my booklet, “Numbers Juggling - Times without the Tables.” Request came from Sol Lederman, who runs the “Wildaboutmath” blog.

I’d heard that name before, but really couldn’t remember much about Sol, so I checked out his blog to see how serious it was.

Wow, it’s a great blog, full of lots of valuable information about math, how to learn and teach math, and the joy of math. You should definitely check it out.

Sol reviewed the booklet, and you can read his review here.

The review was generally positive, but Sol had a very valid and important criticism. Since the greatest value of the booklet is really in the seven follow-up e-mails in the e-mail course, it should be marketed as a course, rather than a booklet.

That got me thinking (as every good book-review should do). So now I am developing real, in-depth, home-study courses for each of the basic operations of arithmetic.

Each will be about thirty modules long. The modules will walk you through the basics to absolutely turbo-charged speed-math methods.

I’ll be telling you more about it as it develops. If you are interested drop me an e-mail. (Use the contact box near the upper right corner of this page).

Now on to the ADD part of this post. Many people who have problems with math have problems with attention, focus, concentration, etc. I am one of them. I have suffered with ADD for as long as I can remember. It was only “officially” diagnosed a few years ago.

As it happens, Sol suffers from it as well. Or suffered. He has a blog dedicated to journaling his recent “cure.” I have not met Sol, and cannot vouch for anything, but he seems very dedicated to describing his experiences honestly.

Let me say that I am a skeptic, down to my bones, and hope you take everything with a grain of salt. But I would investigate what he has to say. I have subscribed to the RSS feed to his site, and intend to look into the methods he as used. You might want to take a look as well.

original photo from Richard Masoner Edited by Brian

Not sure what got me thinking about this today, but I was musing about the commutative property, and how it applies to addition and multiplication. (Yeah, it was a pretty boring morning.)

Specifically, I was thinking about the word, “augend”. The augend of an addition problem is the first of the series of addends. It’s not a word that is usually taught, and I was wondering why not.

You should be aware that addition and multiplication have the commutative property, and subtraction and division don’t. That just means 4+6 is the same as 6+4, and 3*7 is the same as 7*3, but 8-2 and 2-8 are not the same, nor are 9/3 and 3/9.

So it doesn’t really matter which of the numbers is placed where in addition, so you don’t need to specify which is the augend, and which is not. You can call them all addends. Same with multiplication - you can call them all multiplicands. (Not so with minuends, subtrahends, dividends and divisors, though - they don’t commute.)

But I think we should teach about augends. There are several good reasons why, and you can read about them at a lesson I just put up at Names of the numbers in basic arithmetic operations. That lesson is all about why we give the different parts of arithmetic problems their names, (like dividend, divisor, and quotient) and why it makes sense to learn them.

Now that I feel like I’ve cleared this all up for you and me, I’ve got something that I’m not so clear on. Maybe some kind reader has some insight about it she or he’d like to share. It’s this:

Since you must differentiate the names of the parts of subtraction or division problems, what happens if a problem has more than two terms, like 8-3-2? Is there a name for the third term? What if there are fourth or fifth terms?

I assume that they are called, “the second (secondary?) subtrahend, the third (tertiary?) subtrahend, and so on, but I’m not sure.

Anybody got any insights?

You may want to check Names of the numbers in basic arithmetic operations first, though.

By the way, if anybody can write me and tell me why I chose the image that I used for this post, I’ll send them a free Math Mojo e-booklet. (Use the contact me near the top right navigation bar on this page.)

Update: You don’t have to write about that anymore - we have a winner! Mark (see below) got the booklet.

To clarify: The big dog in the picture is “Doggy Daddy,” and the little dog at the door of the train is “Auggie Doggy.” (They are Hannah-Barbera cartoon figures.) They are about to enter a commuter train. Get it? Augend/Auggie, commutative property/commuter train? (Groan.)

I’ve just been perusing a very interesting blog (and a great resource for teachers in public schools). It’s called MathNotations.

This post intrigued and annoyed me, though. (Hey, maybe that’s a sign that it is a good blog!) It’s a poll about which method should be used to teach multidigit multiplication, like 48*73, for example. (If you do go to the link, make sure you scroll down and read the comment on Jan 30th by Michael Paul Goldenberg. It is excellent.)

Unfortunately, this poll is guilty of the same myopia as the American school system in general. It’s about creating a “standard.” Standard is just another word for limitation for people who really don’t know how to excel.

In the case of this poll, it is about choosing (out of an artificially limited group of choices - which is the logical fallacy of “false dichotomies”) how multidigit multiplication should be taught.

The wording of the poll is:

“Here are your options regarding your preference for how multidigit multiplication should be taught in Grades 3-5:”

Um, here are my options? I think not.

One of the great problems in (at least) American education today is that we’re firmly locked, sealed, and vacuum-packed into the box of pedagogical dogma.

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A girl recently asked for some help multiplying.

Professor Homunculus replied:

I have lots of questions to ask you, but first, here is something you can do right away to help you learn multiplication by 2, 3 and 4:

Get out a deck of cards. Make sure they are all (52) there. Now count them by twos. If you end up saying “fifty-two,” and have no cards left, you got it right.

Now try by threes. If you end up saying “Fifty-one”, and have one left over at the end, you got it right again.

Now try by fours. If you end up at “fifty-two” and have none left over, you’re right again.

Now do that over and over. Always count by twos, threes, and fours, no matter what you are counting from now on.

If you get a group of coins, like pennies, as change, count them by threes.

If you have to count the amount of kids in a class, count by threes or twos.

And so on.

Actually counting things in groups is a lot better than looking at tables, and parroting them back.

If you really want a great way to learn to multiply very fast and easily, consider getting a copy of “Numbers Juggling - Times without the Tables.”

You can find a link to it on the right-hand side of each page of these Math Mojo Chronicles.

Also, have you checked out MathMojo.com? Go there and click on the link for “speed multiplication by 11 and 12″. (It’s down the page a bit).

Then, check out:
http://www.squidoo.com/multiplication/
It will teach a cool multiplication trick, but you’ll only be able to do it if you first learn (and practice) what you learn at the “speed multiplication by 11 and 12″ link, above.

You are actually in luck. Recently a new book came out, and I have to say, it is a great book for girls to learn math from.
It’s called “Math Doesn’t Suck,” and it was written by an actress who is also (of all things) a mathematician. It’s a pretty awesome book.

Now for my questions:

How are you at addition? How are you at your other subjects in school? How are you with sports? Do you practice anything regularly (a sport, musical instrument, game?)

When you wrote the comment, were you aware of your misspellings and grammar mistakes? I’m not asking to judge you, just for info for how to help you.

Any answers to those questions will help me figure out how to help you better.

In 2000 an 2001, I was an “expert” on a website that helped kids with math problems. Most of them concerned problems they were having in school.

I’d like to publish some of the exchanges I had with some of the kids. I must mention some “warnings,” though.

1. These questions were asked by real kids, struggling with real problems. They express themselves like real kids. I appreciated that, sometimes. The grammar and spelling is generally miserable. I am going to try not to edit them. I think it’s important if a kid writes, “I want to mulily by too,” that the form of writing gives you several clues to what’s going on. Sometimes.

 

2. The answers were given by a real person (me.) I do not intend to bowlderize anything. There is quite a bit of swearing in them. Why? Because I tend to do a lot of swearing, especially at hypocrisy. Since the questions had a lot to do with schools, there was a lot of hypocrisy to swear at. I’ll try not to go overboard with it, because on this blog and in Math Mojo in general, I try to keep it to a minimum.
   (When we I publish as Joe Archimedes - Hard-boiled Substitute Teacher, the gloves will be coming off, though).

The real reason I am publishing this stuff, is to “speak truth to power.” If administrators feel offended, well…

I feel that there are a lot of parents, teachers, and especially kids out there who have always suspected the things I’m going to say are true, but haven’t heard it expressed. I think it will help to read it.

So here goes nothing…

I’ll be breaking up this post into parts. Here is the first exchange:

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Do you want to understand and be able to multiply in order to:
a) help you (or your child) with life in general and the education that counts, or
b) just help you (or your child) pass the next math test?

If your answer was b, you just saved yourself some time and effort for the next few minutes. You don’t have to read any further. But you’d be costing yourself (or your child) years of frustration. Passing a little (or big, standardized) test is just jumping through an artificial, meaningless, hoop. You don’t have to be a slave to the school system.

If you really need to beat the system, you need to game it. You need to learn math much, much better than they teach it to you in public schools. Then their tests will be a joke, and you will blow them away without being intimidated. But if you just want to learn enough to get through the next test, brother, you are digging your own educational grave.

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I just read an interesting and valuable post by a concerned parent at “Mindless Math Mutterings” (which are anything but). I like that blogger’s thoughtful posts about education.

This particular post was about becoming and expert with math “facts.”

I have one observation that I feel has been terminally left out of this discussion, though:

Although the basic building blocks must be able to be used immediately when needed, what we generically call “memorization” is a dead-end for most people.

We require children to sit and “memorize” in order to learn, but we don’t teach them how to memorize. Memorization is a skill, just like other thinking skills, that needs learning and tweaking. Rote memory is a terrible myth. I mean, it works for some, but it is not the only, nor is it the best, way for most people. (more…)